Emergence of Charge-Imbalanced BCS State and Suppression of Nonequilibrium FFLO State in Asymmetric NSN Junctions

Abstract

We theoretically study nonequilibrium superconductivity in voltage-biased normal metal-superconductor-normal metal (NSN) junctions, focusing on effects of lead-coupling asymmetry and impurity scattering. Using the Keldysh Green's function technique, we extend the thermal-equilibrium mean-field BCS theory to the case where the system is out of equilibrium, to analyze superconducting properties in the nonequilibrium steady state. We find that, in close analogy with the thermal-equilibrium case, the inhomogeneous nonequilibrium Fulde-Ferrell-Larkin-Ovchinnikov (NFFLO) state induced by nonequilibrium electron distributions is highly sensitive to impurity scattering, whereas the uniform nonequilibrium BCS (NBCS) state remains robust against nonmagnetic impurities. Moreover, lead-coupling asymmetry is also found to suppress the NFFLO phase and to split the NBCS phase into two distinct regimes, characterized by the presence or absence of a chemical-potential imbalance between quasiparticles and the condensate. We identify a phase transition or a crossover between these two NBCS states, as well as parameter regimes exhibiting bistability. Our results provide a unified microscopic understanding of nonequilibrium superconductivity in NSN junctions under experimentally relevant conditions and are expected to provide a theoretical framework applicable to a broad class of nonequilibrium superconducting hybrid structures.

Figures (13)

• Figure 1: (a) Model NSN junction Kawamura2024. A thin superconducting layer is sandwiched between two normal-metal leads and is driven out of equilibrium by a bias voltage $V$ applied between the leads. The superconducting order parameter is uniform in the nonequilibrium BCS (NBCS) state, whereas it exhibits a spontaneous spatial modulation in the $x$–$y$ plane in the nonequilibrium FFLO (NFFLO) state. (b) Schematic illustration of the FFLO-type Cooper-pairing mechanism in the nonequilibrium superconductor. The two-step electron distribution in Eq. \ref{['eq.fneq.twostep']} gives rise to two effective Fermi surfaces (FS1 and FS2), leading to Cooper pairs with a nonzero center-of-mass momentum $\bm{Q}$. (c) Schematic phase diagram of a nonequilibrium superconductor in the clean limit with symmetric superconductor–lead couplings Kawamura2024. The system exhibits bistability in the green and blue shaded regions.
• Figure 2: Schematic energy diagram of our model in the nonequilibrium steady state. (a) The energy reference is taken at the Fermi level of the lead 2. The lead 1 is filled up to energy $eV$, and the bias voltage $V$ drives the superconductor out of equilibrium. (b) The energy is measured from the pair chemical potential $\mu_{\rm pair}$, which is defined in Eq. \ref{['eq.OP']}. This choice of energy reference eliminates the trivial time dependence of the superconducting order parameter (see Sec. \ref{['sec.formalism.NBCS']} for details). Unless otherwise stated, results are presented using the energy reference in panel (a), whereas the reference in panel (b) is used in formulating the nonequilibrium BCS theory.
• Figure 3: Calculated superconducting phase transition temperature $T_{\rm env}^{\rm c}$ as a function of bias voltage $V$ in the presence of asymmetry ($P_{\rm lead}>0$) between the two superconductor–lead couplings. The solid and dashed lines represent the transitions to the NBCS ($\bm{Q}=0$) and NFFLO ($\bm{Q}\neq 0$) states, respectively. We set the total lead-coupling strength to $\gamma/\Delta_0=0.1$, and the clean limit $\tau_{\rm imp}=\infty$ is considered. $\Delta_0$ and $T_0^{\rm c}$ denote the superconducting order parameter and critical temperature in the case of thermal-equilibrium superconductivity without coupling to the leads, respectively.
• Figure 4: Cooper pairings associated with the effective Fermi surfaces (FS1 and FS2) induced by the nonequilibrium electron distribution $f_{\rm neq}(\omega)$. In the intra-surface pairing channels (1) and (2), electrons on the same effective Fermi surface form Cooper pairs with zero center-of-mass momentum, resulting in the uniform NBCS state. By contrast, the inter-surface pairing channel (3) yields Cooper pairs with a non-zero center-of-mass momentum, leading to the NFFLO state. The relative electron populations on FS1 and FS2 are approximately given by the lead-coupling ratio $\gamma_1:\gamma_2$.
• Figure 5: Pair chemical potential $\mu_{\rm pair}$ (solid line) and electrostatic potential $\varphi$ (dashed line) along the superconducting phase boundary as functions of the bias voltage $V$. (a) $P_{\rm lead}=0$ (symmetric case). (b) $P_{\rm lead}=0.2$ (asymmetric case). In panel (a), the system is in the NFFLO (NBCS) state when $V > V_{\rm LP}$ ($V < V_{\rm LP}$), where $V_{\rm LP}$ is the voltage of the Lifshitz point shown in Fig. \ref{['Fig.FFLO.asy']}(a).